Differential Forms on Hyperelliptic Curves with Semistable Reduction

Let $C$ be a hyperelliptic curve over a local field $K$ with odd residue characteristic, defined by some affine Weierstrass equation $y^2=f(x)$. We assume that $C$ has semistable reduction and denote by $$\mathcal{X}$ \rightarrow $\text{Spec}$, $\mathcal{O}$K$ its minimal regular model with relative dualizing sheaf ${\omega }_{\mathcal{X}\mathrm{/}{\mathcal{O}}_K}$. We show how to directly read off a basis for $H^0(\mathcal{X},{\omega }_{\mathcal{X}\mathrm{/}{\mathcal{O}}_K})$ from the cluster picture of the roots of $f$. Furthermore we give a formula for the valuation of $\lambda$ such that $\lambda \cdot \frac{dx}{y} \land \dots \land x^{g-1}\frac{dx}{y}$isageneratorfor$\det H^0($\mathcal{X}$,\omega{$\mathcal{X}$/$\mathcal{O}$_K})$.